I just got a new computer that is substantially stronger than my previous one, so I thought it would be able to dump frames better. Unfortunately, that doesn't seem to be the case. The playback seems smoother, but the visuals became garbage. All present codecs seem to be doing this. This is what SUPER SMASH BROS. Melee's title screen ends up looking like:

This didn't happen on my old computer. I also followed the same settings (except for higher resolution.) Could it be my specs?

This has nothing to do with your computer or drivers. This happens if the game's resolution changes while dumping frames, which Melee does automatically. Older versions of Dolphin ignored the game's resolution changes, so this wouldn't get triggered. There is a workaround, which I think is to enable Stretch to Window. You'll probably want to set the window to a proper size first.